At a Harvard workshop on artificial intelligence and math, some researchers said it won’t matter who—human or machine—eventually proves the Riemann hypothesis. But the people actually closest to the problem describe something colder: there’s no clear path forw
For a moment at a Harvard workshop in October 2024. the future of mathematics sounded less like a destiny and more like a vending machine. Mathematicians talked about artificial intelligence the way people talk about a new tool they might use. Most of them, it seemed, weren’t panicking about what AI would do to their field. They were simply curious.
During a coffee break, the conversation tightened around one open problem. In the group I joined, the participants agreed on something that landed with a chill when I heard it: it made no difference whether a human or a computer solved their favorite question. They wanted the proof.
“So you really don’t care whether the Riemann hypothesis gets solved by a human or AI?” I asked. In that circle, it wasn’t just a difference of opinion. It carried a knowing discomfort—smirks, exchanged glances, the sense that the answer might invite consequences.
“An AI that can prove the Riemann hypothesis is not one I’d want to meet,” Andrew Sutherland, a number theorist at the Massachusetts Institute of Technology, said. “If that happens, mathematicians having jobs will be the least of our problems.”
It’s a line that sounds like a joke until you remember how rarely the Riemann hypothesis yields to anything. I’d only been tossing around a name of an open question I’d heard before that day. But the more I returned to it. the more the scale of it became hard to ignore: what kind of mathematical machine—or intellect—could finish this?.
Ever since it was first published in 1859. Bernhard Riemann’s conjecture about prime numbers has sat atop lists of the biggest unsolved mysteries in mathematics. In 1900. David Hilbert drafted a list of problems intended as a blueprint for 20th-century math. and one of them was Riemann’s hypothesis. By the end of the century. the question still wasn’t answered—and it didn’t just remain on the poster. It earned another one.
In 2000, the Clay Mathematics Institute promised a million-dollar bounty to anyone who solved the Riemann hypothesis, placing it among seven “Millennium Problems,” the century’s own aspirational to-do list.
The prize money and the century-long attention can make it seem like momentum should be visible by now. But when mathematicians talk about the problem, the language changes. Alex Kontorovich. a mathematician at Rutgers University. described a flat reality: “The basic status is: nothing is happening. and I don’t really expect anything to happen.” He added that hardly anyone in the field is even working on it. James Maynard. a mathematician at the University of Oxford. put it even more plainly: “I don’t spend too much of my day really thinking about it.” Then. after that pause that suggests frustration more than indifference. he said. “I just don’t really have any good idea of how to get started.”.
That contrast—between the size of the mystery and the smallness of the immediate effort—forces a question that’s hard to shake: why does such a central problem feel, to the people nearest it, so unreachable?
Part of the answer starts with why primes matter at all. Brian Conrad, a mathematician at Stanford University, compared the obsession to one physicists would recognize. “Asking me why number theorists care so much about prime numbers is kind of like asking why physicists care so much about forces. ” he said.
The fixation runs deep into the origin story of math itself, beginning with counting. The ancient Greeks treated whole numbers as paramount. They studied ways of combining them to produce other quantities. But some numbers can’t be built this way—there’s no way to get 17 stones by grouping stones in threes. for example. Those unbuildable building blocks are what the Greeks called prime numbers.
Prime factorization—multiplying a unique combination of primes—became the language for constructing any number that isn’t prime. Around 300 B.C.E., Euclid proved there are infinitely many primes. What no one could explain, for centuries, was why they appear where they do.
Maynard described the strangeness of it in a way that captures the emotional core of long mathematical quests. “On the one hand, this sounds totally bizarre—primes just are what they are. You’re either a prime, or you’re not,” he said. “But one of the best ways to understand prime numbers is often to think about them as being these somewhat random objects.”.
That randomness became a provocation. Mathematicians tried to force meaning out of it by building huge tables of numbers and their prime factorizations—working by tedious. handwritten algorithms. reaching as high as they could. In the late 1700s, Carl Friedrich Gauss turned this into a pastime. In those tables, he saw order where others had only seen disorder.
Between 0 and 100, there are 25 primes. But between 1,000 and 1,100, there are only 16. As numbers rise, primes become rarer. Gauss observed that this trend gets more predictable at higher scales.
The Riemann zeta function enters this story because it describes the locations of primes through a complex mathematical machine. The zeta function defines the sum of an infinite series involving a number with both a real and an imaginary part. Its behavior governs where primes sit.
Gauss compared the numbers of primes in different intervals, going as high as three million. He noticed that the drop-off in the count of primes didn’t just continue—it became more predictable. In the end. he wrote an equation for the trend. predicting roughly how many primes you’d find higher up on the number line.
Gauss didn’t stop there. In an 1801 treatise, he showed primes’ power could be used to perform calculations in finite number systems. But he couldn’t prove the guess about where primes lived.
That effort fell to his disciple: Bernhard Riemann, a young theology student at the University of Göttingen in Germany. Riemann was won over by Gauss’s lectures and trained in the way of numbers. He balked at the primes’ apparent randomness and pushed toward meaning.
He imagined a mathematical machine that could nail down the precise location of every prime. It would walk along the real-number line, picking out primes and skipping the rest. The “workings” of that machine could be described using what became known as the Riemann zeta function.
The zeta function takes a complex number as input—a real component plus an imaginary component. expressed as a coefficient multiplied by the square root of −1. called i. Mathematicians picture it on the complex plane, with real and imaginary axes. Each input point produces a complicated output point. And at special points, the output is zero.
Riemann realized the locations of those zeros mattered. He started with Gauss’s guess for the frequency of primes. which predicts roughly how many primes fall in an interval. But the true primes didn’t match exactly; later mathematicians would prove Gauss’s guess using ideas introduced by Riemann. Even so, Riemann found primes scattered around the prediction.
That leftover error could be described in terms of the zeros of the zeta function. The scattering could be broken into an infinite number of distinct. interacting pieces—like how a musical note decomposes into harmonics. Each zero contributes one harmonic: its imaginary part determines the harmonic’s frequency, its real part the harmonic’s strength.
Finding every zero—determining the pitch and volume of every instrument in this prime symphony—would reveal the precise locations of all primes. fully revealing their “rich music.” But Riemann also knew the solution was beyond reach. Still, anything definitive about the zeros would constrain what primes could do.
In his 1859 paper, Riemann made the crucial guess. The real parts of every zero were all the same: ½. They differed only in their imaginary parts. That property puts them along a single vertical line in the complex plane, the line intersecting the x-axis at ½. The first such point, Riemann predicted, was at 14.13; the next at 21.02.
That is the Riemann hypothesis in its essential form: every zero of the Riemann zeta function lies on this critical line, with real part exactly ½.
Kontorovich described what the conjecture does for understanding primes. “The Riemann hypothesis says all these different instruments in the orchestra of the primes play at exactly the same volume. ” he said. Without it, primes would resemble a conductorless symphony, with instruments sounding at different volumes.
The hypothesis also carries a wonder that isn’t just technical. It entangles a strange function in the complex plane with the fundamental building blocks of whole numbers. It mirrors musical structure. It suggests deeper connections under the surface.
Lauren Williams, a mathematician at Harvard University, explained the feeling that keeps researchers moving through the ambiguity. Her work. she told me. is most exciting “when I discover that two mathematical objects that had no reason to be related to each other actually are. Then it’s kind of a mystery to try to figure out why and how they’re related.”.
The Riemann hypothesis has produced that kind of surprise repeatedly—not only inside mathematics, but beyond it. In 1972. Freeman Dyson. then at the Institute for Advanced Study in Princeton. N.J. had tea with a mathematician who’d noticed strange patterns in the statistics of the zeta function. Dyson recognized the patterns immediately because they matched theory he’d worked out for energy levels of atomic nuclei. He unexpectedly helped solve a pure-math problem. and yet. as Dyson’s story has lingered. “the origin of the connection remains totally obscure.”.
Since then, the Riemann hypothesis has appeared in the random motion of particles, chaos theory, and even the theory of black holes.
And even without a proof in hand, mathematicians haven’t just waited. Kontorovich said there are “hundreds of papers that prove ‘x is true if RH holds. ’” adding. “We’ve already been assuming it was true for a long time.” With the hypothesis as a starting point. number theorists can do far more. The “music” becomes ordered: instead of missing everything, their understanding is missing only the imaginary parts of the zeros.
That is where the mathematics of “L-functions” enters. The Riemann hypothesis inspired analogies across a swath of objects. Many mathematical objects—simple equations to high-dimensional shapes—have associated “L-functions” that describe their properties in the way the zeta function describes primes. Each L-function has its own critical line like x = ½. If its zeros land on that line, the object becomes more coherent.
Mathematicians ran with Riemann’s lead by proposing “generalized Riemann hypotheses.” If all L-functions place their zeros on these natural lines. “the amount of mathematics unlocked is infinitely larger.” So. in a sense. mathematicians already know much of what proving Riemann would unlock—it’s already “in the literature. ” they say. waiting for “this one big missing piece.”.
But the piece still isn’t there. Andrew Granville. a mathematician at the University of Montreal. said. “The Riemann hypothesis has had a few good ideas over the years. but none of them have really gotten to the nut of the matter.” He described the reason with a blunt sense of exhaustion: “They all ran out of steam early.”.
The field’s last major leap has been real, but limited. Two years ago. Maynard of Oxford and M.I.T.’s Larry Guth made the biggest breakthrough on the Riemann hypothesis in decades. All that anyone had proved before was that none of the zeros is too far from the critical line. That boundary had remained stuck for so long that some suspected it couldn’t be pushed further.
Granville said he too had felt that pressure. “I genuinely thought there must be some intrinsic reason that it was stuck,” he said. Maynard and Guth tightened the bound slightly—only slightly. Granville called it marginal and said there was a hard ceiling to it: “These are two of the most brilliant people around. and even they got this marginal improvement.”.
He also said there wasn’t a clear path afterward. “I don’t really view our work as the right direction for solving the Riemann hypothesis,” Maynard said. “I think of our work as more of a workaround for the fact that we don’t know how to solve the Riemann hypothesis.”
That may explain the problem’s peculiar popularity—both high and low. Mathematicians often aim for a sweet spot: hard enough to be worth caring about and easy enough to make some headway with current methods. To Maynard, the lack of any clear route is part of what makes the Riemann hypothesis so important. “I think the Riemann hypothesis is true for a really good reason,” he said. “I just have no idea what that reason is.”.
He added that a proof explaining that reason would bring far-reaching insights and a new level of command over numbers. “It’s the veiled contents of this mythical proof that so beguile mathematicians they would cede their autonomy to a large language model to see them.”
Maynard ended on a line that crystallizes both the hunger and the humility behind the missing proof. “Provided I could understand the proof, be it from an alien or God or AI,” he said, “I would be superexcited.”
Riemann hypothesis prime numbers Riemann zeta function Clay Mathematics Institute Millennium Problems artificial intelligence Andrew Sutherland Alex Kontorovich James Maynard Bernhard Riemann David Hilbert